Mercury has an orbital eccentricity of 0.21. Find the perihelion and aphelion distances of Mercury. What is the ratio of Mercury's orbital speed at perihelion to that at aphelion?

To find the perihelion and aphelion distances of Mercury, we need to know the length of its semi-major axis (a) and its eccentricity (e).

The formula to calculate the perihelion and aphelion distances is as follows:

Perihelion distance = a * (1 - e)
Aphelion distance = a * (1 + e)

Given that the eccentricity of Mercury's orbit (e) is 0.21, we need to determine the semi-major axis (a). The semi-major axis represents the average distance from the center of Mercury's elliptical orbit to the Sun.

The average distance from Mercury to the Sun is approximately 57.9 million kilometers (36 million miles). This value is equal to Mercury's semi-major axis (a).

Now we can calculate the perihelion and aphelion distances:

Perihelion distance = a * (1 - e) = 57.9 * (1 - 0.21) = 45.681 million kilometers (28.361 million miles)

Aphelion distance = a * (1 + e) = 57.9 * (1 + 0.21) = 70.019 million kilometers (43.496 million miles)

Therefore, the perihelion distance of Mercury is approximately 45.681 million kilometers (28.361 million miles), and the aphelion distance is approximately 70.019 million kilometers (43.496 million miles).

To calculate the ratio of Mercury's orbital speed at perihelion to that at aphelion, we use Kepler's second law, which states that an imaginary line joining a planet to the Sun sweeps out equal areas in equal intervals of time.

Since Mercury moves faster when it is closer to the Sun (perihelion), the ratio of the orbital speeds at perihelion and aphelion can be determined using the conservation of angular momentum:

Vp / Va = Ra / Rp

where Vp is the orbital speed at perihelion, Va is the orbital speed at aphelion, Rp is the perihelion distance, and Ra is the aphelion distance.

Using the calculated distances above:

Vp / Va = 70.019 / 45.681 = 1.5306

Hence, the ratio of Mercury's orbital speed at perihelion to that at aphelion is approximately 1.5306.